Toeplitz matrix

$ T $- matrix

An infinite matrix $ ( a _ {nk} ) $, $ n, k = 1, 2 \dots $ satisfying the conditions:

$$ \sum _ {k = 1 } ^ \infty | a _ {nk} | \leq M,\ \ n = 1, 2 \dots $$

where $ M $ does not depend on $ n $;

$$ \lim\limits _ {n \rightarrow \infty } \ a _ {nk} = 0,\ \ k = 1, 2 , . . . ; $$

$$ \lim\limits _ {n \rightarrow \infty } \sum _ {k = 1 } ^ \infty a _ {nk} = 1. $$

These conditions are necessary and sufficient for regularity (cf. Regular summation methods) of the matrix summation method defined by sending a sequence $ \{ s _ {n} \} $ to a sequence $ \{ \sigma _ {n} \} $ via the matrix $ ( a _ {nk} ) $:

$$ \sigma _ {n} = \ \sum _ {k = 1 } ^ \infty a _ {nk} s _ {k} . $$

The necessity and sufficiency of these conditions for regularity were proved by O. Toeplitz in the case of triangular matrices.

References

[1]	O. Toeplitz, Prace Mat. Fiz. , 22 (1911) pp. 113–119
[2]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[3]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)

Comments

In the literature the term "Toeplitz matrix" is also used for (finite or infinite) matrices $ ( a _ {jk} ) $ which have the property that $ a _ {jk} $ depends on the difference $ j- k $ only, i.e., $ a _ {jk} = \alpha _ {j-k} $ for all $ j $ and $ k $. The material below concerns Toeplitz matrices in this sense.

Finite Toeplitz matrices have important applications in statistics, signal processing and systems theory. For such matrices there are different algorithms (N. Levison, I. Schur and others) for inversion. The inverse of a finite Toeplitz matrix $ A = ( \alpha _ {j-k} ) _ {j, k= 1 } ^ {n} $ is not Toeplitz, but it is of the following form:

$$ \tag{a1 } A ^ {-1} = $$

$$ = \ x _ {0} ^ {-1} \left \{ \left ( \begin{array}{cccc} x _ {0} & 0 &\dots & 0 \\ x _ {1} &x _ {0} &\dots & 0 \\ \cdot &\cdot &\dots &\cdot \\ x _ {n} &x _ {n-1} &\dots &x _ {0} \\ \end{array} \right ) \left ( \begin{array}{cccc} y _ {0} &y _ {-1} &\dots &y _ {-n} \\ 0 &y _ {0} &\dots &y _ {-n+ 1} \\ \cdot &\cdot &\dots &\cdot \\ 0 & 0 &\dots &y _ {0} \\ \end{array} \right ) \right . - $$

$$ - \left . \left ( \begin{array}{cccccc} 0 & 0 & 0 &\dots & 0 & 0 \\ y _ {-n} & 0 & 0 &\dots & 0 & 0 \\ y _ {-n+ 1} &y _ {-n} & 0 &\dots & 0 & 0 \\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot \\ y _ {-1} &y _ {-2} &y _ {-3} &\dots &y _ {-n} & 0 \\ \end{array} \right ) \left ( \begin{array}{ccccc} 0 &x _ {n} &x _ {n-1} &\dots &x _ {1} \\ 0 & 0 &x _ {n} &\dots &x _ {2} \\ \cdot &\cdot &\cdot &\dots &\cdot \\ 0 & 0 & 0 &\dots &x _ {n} \\ 0 & 0 & 0 &\dots & 0 \\ \end{array} \right ) \right \} , $$

where $ x _ {0} $ is assumed to be $ \not\equiv 0 $, and $ x _ {0} \dots x _ {n} $ and $ y _ {-n} \dots y _ {0} $ are solutions of the following equations:

$$ \sum _ { k= } 0 ^ { n } \alpha _ {j-k} x _ {k} = \ \delta _ {j0} ,\ \ \sum _ { k= } 0 ^ { n } \alpha _ {j-k} y _ {k-n} = \ \delta _ {jn} \ ( j = 0 \dots n ) . $$

Here $ \delta _ {jk} $ is the Kronecker delta. Formula (a1) is known as the Gohberg–Semencul formula (see [a4]). See [a5], [a6] for further development in this direction.

Infinite Toeplitz matrices $ ( \alpha _ {j-k} ) _ {j, k= 1 } ^ \infty $ define an important class of operators on the Hilbert space $ l _ {2} $ which may be analyzed in terms of their symbol $ \sum _ {j=- \infty } ^ \infty \alpha _ {j} \lambda ^ {j} $, $ | \lambda | = 1 $. The theory of these operators is rich and contains inversion theorems (based on factorization of the symbol), Fredholm theorems, explicit formulas for the index in terms of the winding number of the symbol, asymptotic formulas for the determinant of its finite sections, etc. In fact, the infinite Toeplitz matrices form one of the few classes of operators for which explicit inversion formulas are known and they provide one of the first examples of the modern index theory. For the recent literature see [a2], [a3], [a7]. Infinite Toeplitz matrices with matrix entries of which the symbol is rational are of particular interest, and the corresponding operators may be analyzed in terms of methods from mathematical system theory (see [a1]).

References

[a1]	H. Bart, I. Gohberg, M.A. Kaashoek, "Wiener–Hopf integral equations, Toeplitz matrices and linear systems" I. Gohberg (ed.) , Toeplitz Centennial , Birkhäuser (1982) pp. 85–135
[a2]	A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990)
[a3]	I.C. [I.Ts. Gokhberg] Gohberg, I.A. Feld'man, "Convolution equations and projection methods for their solution" , Transl. Math. Monogr. , 41 , Amer. Math. Soc. (1974) (Translated from Russian)
[a4]	I. [I.Ts. Gokhberg] Gohberg, A.A. Semencul, "On inversion of finite-section Toeplitz matrices and their continuous analogues" Mat. Issled. Kishinev , 7 : 2 (1972) pp. 201–224 (In Russian)
[a5]	G. Heinig, K. Rost, "Algebraic methods for Toeplitz-like matrices and operators" , Akademie Verlag (1984)
[a6]	T. Kailath, J. Chun, "Generalized Gohberg–Semencul formulas for matrix inversion" H. Dym (ed.) S. Goldberg (ed.) M.A. Kaashoek (ed.) P. Lancaster (ed.) , The Gohberg Anniversary Collection , I , Birkhäuser (1989) pp. 231–246
[a7]	N.K. Nikolskii (ed.) , Toeplitz operators and spectral function theory , Birkhäuser (1989) (Translated from Russian)